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AN  INTRODUCTION 


TO  THE 


Differential  Calculus 


BY  MEANS  OF  FINITE  DIFFERENCES 


By    ROBERDEAU    BUCHANAN,    S.  B., 

Assistant  in  the  Nautical  Almanac  Office,  U.  S,  Naval  Observatory, 

Author  of  the  Mathematical  Theory  of  Eclipses, 

Treatise  ^n  the  Projection  of  the  Sphere, 

Etc. 


Reiirinted  from  Popular  Astronomy,  Vol.  XIII,  Nos.  5  and  6. 


AN  INTRODUCTION 


TO  THE 


Differential  Calculus 


BY  MEANS  OF  FINITE  DIFFERENCES 


By    ROBERDEAU    BUCHANAN,    S.  B., 

Assistant  in  the  Nautical  Almanac  Office,  U.  S.  Naval  Observatory, 

Author  of  the  Mathematical  Theory  of  Eclipses, 

Treatise  on  the  Projection  of  the  Sphere, 

Etc. 


-    \  e  «   A  ^  )r 

^  or  THt 

VNlVERSiTY 

CALIFOHiii> 


1905 


Reprinted  from  Popular  Astronomy,  Vol.  XIII,  Nos.  5  and  6. 


OA  3  o  Co 
33^ 


Entered  according  to  Act  of  Congress  in  the  year  1905 

'By   ROBERDEAU  BUCHANAN 
In  the  Office  of  the  Librarian  of  Congress  at  Washington 


tNIV£RS<TY 


or 


PREFACE, 


AN  INTRODUCTION  TO    THE   DIFFERENTIAL,  CALCULUS, 
BY  MEANS  OF  FINITE  DIFFERENCES. 


BY  ROBERDEAU  BUCHANAN.  S.  B. 


From  the  time  the  author  commenced  the  study  of  the  Calculus 
he  has  been  of  the  opinion  that  an  introductory  text-book  was  a 
want  to  be  supplied.  And  Finite  Differences  undoubtedly  offer 
the  best  method  of  explanation.  All  that  the  student  is  required 
to  know  at  first,  is  mere  differencing  which  is  easily  understood. 
In  article  19  and  those  following,  formulae  from  Finite  Differences 
are  given,  but  not  their  derivation,  and  the  student  can  regard 
them  and  use  them  simply  as  algebraic  equations.  They  con- 
stitute the  proof  of  this  method  of  explaining  the  Calculus  and 
the  student  may  omit  them  until  he  advances  further  in  the 
study. 

Professor  Newcomb  says  in  the  Preface  to  his  Algebra  "that 
all  mathematical  conceptions  require  time  to  become  engrafted 
upon  the  mind,  and  the  more  time,  the  more  their  abstruseness." 
And  again  in  the  same  work,  "one  well  known  principle  underly- 
ing the  acquisition  of  all  knowledge  is  that  an  idea  cannot  be 
fully  grasped  by  the  youthful  mind,  unless  it  is  presented  in  some 
concrete  form.  Whenever  possible  an  abstract  idea  must  be  em- 
bodied in  some  visible  representation." 

For  the  first  reason,  the  author  has  presented  the  first  example 
several  times,  varying  the  questions  each  time;  and  for  the  second 
reason  has  explained  the  example  in  more  than  one  way — by 
means  of  numbers,  by  F'inite  Differences,  by  an  algebraic  equa- 
tion, by  a  differential  equation,  and  by  geometrical  representa- 
tion; all  converging  to  the  same  end,  that  the  student  may  see 
the  similarity  between  them  all  and  thus  gain  his  idea  of  a  differ- 
ential, and  also  of  the  first  principles  of  the  calculus.  It  is  the 
purpose  of  the  author  to  give  only  the  elementary  idea  leaving 
the  rest  to  the  student's  instructor. 

Since  reading  Professor  Bledsoe's  Philosophy  of  Mathematics^ 
some  years  ago,  the  author  has  not  been  satisfied  with  the  ex- 

140680 


planations  usually  given  of  the  calculus  which  make  a  finite 
difference  become  a  differential  by  suppressing  certain  terms; 
whereas  a  finite  difference  cannot  become  a  differential  except  by 
a  change  of  value  to  which  these  suppressed  terms  contribute. 
The  author  does  not  wish  to  be  understood  as  denying  the  truths 
of  the  Calculus,  but  suggests  that  a  better  method  of  explana- 
tion might  be  adopted.  Nor  does  he  wish  to  hurt  the  feelings  of 
writers  on  the  calculus;  and  for  that  reason  has  refrained  from 
quoting  the  works  of  any  living  author.  It  is  the  method  of  ex- 
planation which  he  controverts,  which  seems  to  have  been  al- 
most unanimously  adopted  by  writers  since  the  time  of  Euler  in 
1787,  and  perhaps  earlier.  THE  AUTHOR. 

2015  Q  Street,  Washington,  D.  C. 
1905,  February  27. 


PRINCIPAL  CONTENTS, 


Art.     1.    Method  of  explanation  by  the  DoQtrine  of  Limits. 

3.    The    present    method  by  Finite  Differences  ex- 
plained.   Elementarj^  differentiation 7 

The  effect  of  constants 8 

5.  Objections  stated  to  the  method  by  Limits 11 

6.  Method  by  Finite  Differences  further  explained; 

Graphic  representation;  The  effect  of  con- 
stants; Explanation  of  two  forms  arising 
from  integration 14 

16.    Special  cases  explained;  Maxima    and   Minima, 

Points  of  Inflexion;  Cusps 19 

19.  Rigorous  demonstration  of  the  method  by  Finite 
Differences.  Differential  coefficients  in  terms 
of  Finite  Differences.  The  distinction  one  of 
Position 21 

22.    Further  criticisms  on  the  Method  by  Limits 23 

25.    The  reverse  problem,   Finite  Differences  derived 

from  Differential  Coefficients 25 

28.    Concluding  remarks  on  the  Method  by  Limits....        27 


8  An  Introduction  to  the  Differential  Calculus. 

Then,  h  being  infinitely  small  compared  with  2x  can  be  neglected. 
Others  say  that  here  h  may  undergo  any  change  of  value  with- 
out affecting  x,  so  letting  h  diminish  until  it  is  zero  we  have  equa- 
tion (5)  as  given  above.* 

In  Davis  and  Peck's  Dictionary  of  Mathematics,  Article  In- 
ffnitesimal,  this  principle  is  very  broadly  stated. 

''When  several  quantities  either  finite  or  infinitesimal  are  con- 
nected together  by  the  signs  plus  and  minus,  all  except  those  of 
the  lowest  order  may  be  neglected  without  affecting  the  value  of 
the  expression.    Thus 

or  [  (8) 

dx  +  dx^  -irdx^  =  dx''\ 

2.  Professor  Bledsoe  in  his  Philosophy  of  Mathemicsf  attacks 
this  method  of  explanation,  because  in  Equation  7,  h  is  supposed 
to  be  equal  to  zero  on  one  side  of  the  equation  but  retained  as  a 
small  finite  quantity  on  the  other.  And  also  in  equations  (8) 
asks,  Can  the  first  one  be  exactly  equal  to  a?  (p.  64)  And  also 
"No  one  can  look  the  principle  fairly  and  fully  in  the  face,  that  an 
infinitely  small  quantity  may  be  subtracted  from  a  finite  quantity, 
without  making  .even  an  infinitely  small  difference  in  its  value, 
and  yet  regard  it  otherwise  than  absurd"  (ibid  p.  63)  Professor 
Bledsoe  states  that  this  method  is  sanctioned  by  such  names  as 
Roberval,  P?iscal,  Leibnitz,  the  Marquis  de  L'Hopital  and  others, 
and  yet  it  is  apt  to  inspire  the  student,  that  the  calculus  is 
''merely  a  method  of  approximation."     (pp.  65,  70). 

DeMorgan,  a  great  name  in  mathematics  has  evidently  himself 
seen  the  objections  to  this  method  and  gives  a  long  explanation 
of  a  limit  (which  however  he  makes  no  further  use  of);  then  steps 
over  the  difficulty  bj'  giving  a  number  of  differentials  "without 
demonstrating  them,  therefor,  or  even  defining  them."t 

Newton,  one  of  the  discoverers  of  the  calculus  conceived  a  line 
to  be  made  by  the  motion  of  a  point,  its  path,  in  fact;  hence  the 
names  he  adopts— fluents  and  fluxions. 

Professors  Rice  and  Johnson  of  the  United  States  Naval  Acad- 
emy have  published  a  calculus  founded  on  the  idea  of  rates  and 
velocities,  etc.  The  works  of  these  three  latter  writers  are  not 
open  to  the  objections  which  have  been  made  to  the  method  by 
limits. 

3.  Author's  explanation  by  means  of  Finite  Differences.    On 

*  James  Haddon  Examples  and  Solutions  in  the  Differential  Calculus.    Weale's 
Series. 

t  Published  by  J.  B.  Lippincott  &  Co.,  Philadelphia  1874,  p.  212. 
t  De  Morgan  Differential  Calculus,  London  1842;  chapter  ;. 


Finite  Differences.  9 


account  of  their  analogy,  one  to  the  other,  Finite  Differences 
seem  to  be  the  natural  explanation  to  the  Calculus,  when  their 
diversity  is  also  pointed  out.  This  method  is  more  general  than 
those  mentioned  in  the  foregoing  pages.  Moreover  it  easily  ex- 
plains and  verifies  the  objection  raised  by  Professor  Bledsoe. 

It  will  be  sufficient  for  our  purpose  at  present  to  assume  as  a 
general  equation,  the  following: 

17  =  a  +  (6  +  cx)2  (9) 

First,  let  a  =  0  6  =  0  and  c  —  \  which  gives  the  equation 
(1)  of  Article  1. 

W  =  a2  (10) 

Now  assume  successive  values  of  x,  the  independent  variable, 
positive  and  negative  and  find  the  resulting  values  of  u.  Place 
the  values  of  x  in  column  1,  Example  1,  and  the  resulting  values 

Example  1,  Equation  10.    u  =  x^ 


Col.  1. 

Col.  2. 

Col.  3. 

Col.  4. 

Col.  5. 

X 

u 

Ai 

A2 

A3 

—5 

+25 

-9 

•  4 

16 

7 

+2 

0 

3 

9 

5 

2 

0 

2 

4 

3 

2 

0 

-1 

+   1 

—1 

2 

0 

0 

0 

+  1 

2 

0 

+1 

+  1 

3 

2 

0 

2 

4 

5 

2 

0 

3 

9 

7 

2 

0 

4 

16 

9 

2 

0 

'5 

25 

+11 

+  2 

+6 

+36 

of  u  in  column  2,  which  is  headed  u.  The  proper  signs  of  the 
quantities  should  be  carefully  considered  and  written  whenever  a 
sign  changes.  Next,  form  the  column  of  First  Differences,  A^,* 
(column  3)  by  subtracting  algebraically  each  number  from  the 
number  below  it,  in  column  2  and  placing  the  difference  below 
the  line,  that  is,  0  opposite  the  space  between  the  two  numbers.  It 
will  be  seen  presently  that  this  position  for  the  difference  is  the 


*  The  character  A  in  the  Calculus  of  Finite  Differences  is  a  symbol  of  opera- 
tion as  d  is  in  the  Differential  Calculus,  it  is  often  however  taken  to  represent  the 
Differences  themselves;  in  the  following  examples  it  has  this  latter  meaning. 


10  An  Introduction  to  the  Differential  Calculus. 

correct  place,  and  it  is  important  that  it  should  be  so  placed.* 
This  column  is  headed  Aj,  the  usual  designation  for  a  first  differ- 
ence. Next  in  like  manner  form  the  column  oi  Second  Differences 
Ag  (column  i)  subtracting  algebraically  as  before,  each  number 
in  column  3  from  the  one  below  it,  and  placing  the  difference  on 
the  line,  that  is,  opposite  the  space  between  the  two.  This  is  the 
proper  position  for  the  second  difference.  This  column  is  headed 
Ag  and  in  the  present  example  is  a  constant  2.  There  are  no  Third 
Differences  A^  in  this  example  since  if  we  try  to  subtract  our  re- 
sults equal  zero  and  moreover  all  the  higher  differences  must  be 
zero  also.  The  second  differences  being  a  constant  are  independent 
of  the  value  of  the  function  u. 

4.  The  reader  will  now  have  but  little  difficulty  in  understand- 
ing the  meanings  of  differential,  and  differential  coefficient: 

The  differences  of  the  values  of  x  column  1  which  are  all  unity 
are  practically  dx  the  differential  of  x;  and  the  numbers  column 
3,  the  First  Differences,  are  practically  the  first  differential  co- 
efficient of  the  values  of  u  in  column  2,  depending  upon  the 
values  of  x,  column  1.  And  the  Second  Differences  column  4  are 
practically  the  second  differential  coefficient  of  u.  These  numbers 
show  the  analogy  of  differences  and  differentials.  The  word 
practically  used  above  alludes  to  their  diversity  which  will  be 
considered  presently. 

To  show  these  statements,  resume  equation  (10) 

u  =  x'    1 

da  \  (11) 

Differentiate      -^  =  2x 

This  equation  will  practically,  that  is  very  nearly,  give  the 
numbers  in  column  3,  the  First  Differences.  •  But  first  an  im- 
portant distinction  between  a  difference  and  a  differential  must 
be  stated.  The  value  +  5  is  the  first  difference  belonging  to  the 
value  ofu  =  +  4?orx=2  and  properly  is  placed  below  the  line. 
The  differential  coefficient  of  x  =  -}-  2,  and  u  =  +  4  would  prop- 
erly stand  on  the  line  with  u,  so  that  its  value  would  be  between 

3  and  5.    Now  by  equation  (11)  compute  various  values   of  -i- 

and  compare  them  with  the  example.  They  are,  in  equations  of 
the  second  degree,  correctly  speaking,  the  means  of  the  two 
differences  above  and  below  the  line,  thus: 


*  In  works  on  Finite  Differences  the  line  of  Differences  is  usually  placed  on  the 
line  of  the  primitive;  or  else  the  values  of  the  function  are  placed  on  a  line  with 
the  differences  in  columns  below.  The  method  described  in  the  text  is  the  more 
general  and  seems  to  possess  more  advantages  than  any  other. 


(: 


or   THt 

UNIVERSITY 

OF 


Objections  to  the  Method  by  Limits.  11 

^" 
If  A  =  —  4  ,  ^  =  2.Y  =  —  8       0/3  the  line 

du 
X  =  —  1  -7—  =:  2a:  =  —  2       on  the  line 

du 
A'  =  +  2  d^  ~  ^'^'  —  +  ^       '^"  ^^^  ^^"^ 

du 
X  —  -\-  2.  5       below  the  line  ^  =  2-y  =  -f  5       below  the  line 

It  should  be  remarked  that  for  so  low  an  order  as  the  square 
the  differentials  and  differences  agree  with  one  another  more 
closely  than  they  do  in  the  higher  powers,  which  will  be  shown 
by  an  equation  of  the  fifth  power  on  a  subsequent  page. 

Resume  equation  (4)  of  Article  1. 

m  —  v  =  2xh  +  A2 
in  which  h  is  the  increment  of  x  and  is  always  taken  as   unity  in 
Finite  Differences.     It  is  also  so  taken   in   Example  1,  column  1. 
Hence  we  have 

ui  —  u  =  2x  +  1  (12) 

from  which  we  can  compute  the  first  differences,   remembering 
that  they  fall  below  the  line,  and  we  have  as  follows: 

If  A'  =  —  4,      wi  —  u  =  2x  4-  1  =  —  8  +  1  =  —  7 

A-  =  —  1,        171  —  t/  =  2A+  1  =  — 2+1  =  — 1 

A  =  +  2,       ui  —  u  =-  2a  +  1  =  4-  4  +  1  =r  +  5 
A  =  -f  2.  5,  ui  —  u  =  2a  +  1  =  +  5  +  1  =  +  6 

The  last  value  -+-  6  should  fall  half  a  line  below  the  position  ot 
X  =  2.5  which  brings  it  upon  the  line  below.  We  see  here  that 
universally  for  any  given  value  of  x  the  differences  and  differen- 
tials have  not  the  same   values. 

Differentiate  equation   (11)  a  second  time 

a  constant  which  is  the  value  in  Example  1,  column  4  of  the 
finite  differences. 

5.  Statement  of  the  objections  to  the  usual  Method  by  Limits. 
Equation  (4)  is  a  correct  equation  in  Finite  Differences  provided 
/?  is  a  constant,  usually  taken  as  unity.  With  this  value  the 
equation  gives 

ui  —  u  =  2a  -h  1 

Butas,Equa.  (11)        ^        =2a 

We  have  -^.     =(ui  — a)  — l 

from  which  the  differential  coefficient  of  equation  (10)  may  be 
computed  from  the  finite  differences,  as  we  have  shown  above. 
We  see  also  that  to  form  the  differential  coefficient  the  quantity 


12 


An  Introduction  to  the  Differential  Calculus. 


h^  or  unity  disappears  by  numerical  subtraction  merging  into  the 
term  {u^  —  u).  By  the  method  of  limits  this  quantity  h  is  divided 


Fig.l. 


5      -\     4      -2      4/tf  VI      +?*3 


1   /■  / 

/    1   / 

!     /    !  ' 

/     '  -^ 

'      '/ 

/      / 

'  '     / 

/     / 

'/ 

\ 

\ 
\ 

\ 

out  in  order  to  get  rid  of  it  as  a  factor  to  2x  giving  the  equation 
7  as  follov^s: 


t-x 


+4 


Equation  7, 


2a-  +  h 


in  v^hich /2  in  the  second  member  decreases  until  it  disappears 
while  in  the  first  member  it  is  retained  as  an  infinitely  small 
quantity  then  called  dx.  This  is  by  far  the  most  unmathematical 
feature  of  this  method.  There  is  also  another  point.  The 
equivalent  of 

du         .  ,  wi  —  u 

i«  -nriT 

dx 


IS  not 


but; 


Ul  —  u 


Objections  to  the  Method  by  Limits.  13 

Another  objection  noticed,  is  that  equation  (7)  above  given,  is 
a  correct  equation  in  Finite  Differences  if  h  is  taken  equal  to 
unity,  which  gives 

ui  —  u  =  2x  4-  1 

but  it  would  hardly  be  suitable  to  let  this  term  decrease  until  we 
have  1  =  0!  so  the  notation  h  is  substituted.  This  constant 
unity  is  the  foundation  of  the  theory  of  Finite  Differences.  With 
the  quantity  /?,  the  above  equation  becomes  equivocal.  When  h 
is  unity  or  finite  the  equation  is  one  of  Finite  Differences;  but 
when  h  is  infinitely  small  or  zero  it  is  supposed  to  be  a  differential 
equation. 

The  doctrine  of  limits  cannot  in  this  manner  be  applied  to  an 
equation  of  Finite  Differences  in  which  the  whole  essence  and  life 
of  the  equation  depends  upon  this  quantity  h  being  a  constant 
and  a  £nite  quantity.  In  order  to  find  the  differential  coefficient 
from  the  finite  difference  this  method  makes  the  constant  h  dis- 
appear by  division;  the  author's  method  on  the  other  hand 
makes  it  disappear  by  subtraction  which  leaves  the  absolute 
zero.  This  subject  will  be  still  further  examined  on  a  subsequent 
page. 


14  An  Introduction  to  the  Differential  Calculus. 


II. 

6.  Graphical  Representation.  In  Fig.  1  draw  the  axis  of  X  a 
horizontal  line,  and  the  axis  of  U  perpendicular  to  it  at  the  point 
O.  Lay  off  the  negative  values  of  x  on  the  left  and  the  positive 
on  the  right.  The  spaces  betv^een  these  points  are  each  dx  vyrhich 
is  taken  as  unity  in  the  example  column  1.  Upon  these  points 
erect  the  ordinates  and  lay  off  the  values' of  u  resulting  from  the 
values  of  x  in  example  1.  These  latter  points  a  b  c  etc.  may  be 
connected  by  a  curve  v^hich  represents  the  equation  10  or  11. 

7.  If  from  the  points  a  b  c,  etc.,  already  found  we  draw  the 
horizontal  lines  aa',  bb',  cc',  etc.,  to  the  next  ordinate  on  the  right 
side  (which  is  the  positive  direction  of  the  motion  of  x,  from  left 
to  right,)  the  lines  o'a  a'b  b'c,  etc.,  are  the  finite  differences,  be- 
longing to  the  values  of  u.  These  lines  form  a  familiar  figure  in 
most  works  on  the  calculus,  but  which  is  usually  not  sufficiently 
explained,  beyond  a  mere  statement.  The  angles  which  the 
chords  Oa  ab  be,  etc.,  make  with  the  axis  of  X  are  expressed  by 
the  tangents 

ao'  ba'  cb'  /\u 


etc.  = 


o'O        '  a' a  b'b         '    '^'-'-  ~    A^ 

which  are  very  nearly  the  tangents  to  the  middle  points  of  the 
curve;  but  not  exactly  because  the  curvature  is  different  at  the 
two  ends  of  the  chords.  The  differential  coefficient  for  an  or- 
dinate midway  between  those  in  the  figure  takes  account  of  the 
curvature,  and  gives  the  tangents  to  these  middle  points.  It 
will  be  noticed  that  on  the  left  of  the  axis  of  U,  du  is  laid  off 
downwards,  or  in  a  negative  direction,  so  that  the  tangent  is  a 
negative  quantity,   showing    that  the*  curve  is   moving  in  the. 

d^u 
negative  direction,  or  downwards,  in  the  figure.    And  —z-^  is  pos- 
itive; which  also  shows  that  the  curve  is  convex  towards  the 
axis  of  X,  Fig.  1. 

8.    If  in  Equation  (10)  we  give  the  negative  sign  to  the  right 
member 

U      =— A'2  (13) 


we  have 

du 

dx 
d^u 
dx' 


2x 

(14) 
=  —2 


( 


'^^    TH'. 

UNIVERSl 


Constant  Disappears  in  Differentiation.  15 


Which  is  shown  as  follow  s, 

Example  1,  biz: 

.Y 

u 

Ai 

A 

-2 

—4 

+3 

—1 

—1 

+  1 

-2 

0 

0 

—1 

2 

+1 

—1 

3 

2 

2 

4 

2 

4         16  —2 

—9 
+  5      —25 

And  comparing  with  Example  1,  it  is  seen  that  the  effect  is  to 
change  the  signs  of  all  the  difference.  The  second  differences  be- 
ing negative,  show  that  the  curve  has  a  maximum  point  and 
that  it  is  concave  to  the  axis  of  x  (that  is  concave  to  a  line  be- 
low the  curve);  and  graphically,  Fig.  1,  it  has  been  revolved 
round  the  axis  of  X  180°  and  lies  wholly  below  it.  The  effect 
upon  the  Finite  Differences  is  also  seen.  For  x  =  -{-  S  u  becomes 
—  9,  and  the  first  difference  (below  the  line  as  usual,)  is  —  7. 

9.  A  constant  connected  with  a  variable  by  the  sign  plus  or 
minus  disappears  in  differentiation,  but  the  curve  is  moved  along 
the  axis  of  U.  In  the  general  equation  (9),  let  a  —  2  aconstant, 
thus  we  have 

u  =2  +  A-2 


da 

-r-=:2x 

ox 


(15) 


With  this  equation  form  the  finite  differences  as  in   Example  2. 
It  is  seen  that  although  the  column  u  is  not  the  same  as  in   Ex- 


\.MPLE  2,       az 

=  2  +  ^ 

2        (15) 

X               u 

Ai 

A2 

—5        +27 

—9 

4           18 

7 

+2 

3            11 

5 

2 

2             6 

3 

2 

— 1     '        3 

—1 

2 

0             2 

+  1 

2 

+  1              3 

3 

2 

2              6 

5 

+  2 

+  3       +11 

16  An  Introduction  to  the  Differential  Calculus: 

ample  1,  yet  the  differences  are  identical,  since  the  constant  can- 
cels out  in  the  subtraction.  In  other  words  we  have  a  portion 
of  the  example  as  follows: 


X  =  —  5, 

n  =  4-  25  +  2 

x  =  — 4, 

M=  +  16  +  2 

x  =  —  3, 

«=+    9-1-2 

when  u  is  seen  to  be  composed  of  two  terms  the  first  being  iden- 
tical with  Example  1,  and  the  other  a  constant.  The  first  term 
gives  of  course  the  same  differences  as  in  Example  1  but  the 
second  term  has  no  differences.  That  is  a  constant  term  has  no 
differential  and  disappears  in  differentiation  as  it  does  in  Finite 
Differences.  If  we  plot  the  numbers  in  Example  2  on  Fig.  1,  we 
find  that  it  is  similar  to  that  figure,  but  raised  two  units  along 
the  axis  of  U  or  what  is  the  same  thing  it  will  be  the  curve  of 
Example  1  if  the  axis  of  X  should  be  drawn  through  the  point  P. 
The  effect  of  this  independent  constant  is  thus  not  to  alter  the 
curve  itself,  but  to  place  it  differently  upon  the  axes. 

10.  We  see  also  that  when  integrating  we  must  always  afl&x 
the  letter  C  to  the  result  to  allow  for  a  constant  that  may  ha,ve 
disappeared  in  differentiating. 

11.  Effect  of  a  constant  connected  with  the  function  of  the 
variable.  The  curve  moved  along  the  axis  of  X.  In  the  general 
equation  9,\ct  b  =  -\-  2 

Then 

+  4x  +  4  I 

[  (16) 

This  equation  with  its  difterences  is  shown  in  Example  3,  where 
u  is  seen  to  differ  from  Example  1  by  a  variable  quantity,  but  the 

Example  3,    w  =  (2  -f  x^       (16) 

Ai       As 


+  2 
3 

2 
— 1 

2 
+1 


u 

=--(x  +  2)^ 

da 

=  2x  -f  4 

A- 

u 

■5 

+9 

4 

4 

3 

+1 

2 

0 

-1    +1 


+1  9  +2 

+2      -hl6 

differences  are  still  the  same  indicating  the  same  curve  as  before; 


Constant  Factor  Appears  in  Differences.  17 

but  still  differently  placed  upon  the  axes,   that  is,   moved  along 
the  axis  of  X,  two  units. 

12.  Explanation  of  two  forms  of  an  equation  resulting  from 
Integration,  differing  by  a  constant.  If  we  Integrate  equation 
(16)  viz. 

du 

we  have 

u  =  J{2x  +  4)  dx  =  x2  +  4x  +  C  (17) 

We  cannot  determine  the  value  of  the  constant  C  from  this 
equation  alone,  for  if  x  =  0,  C  =  0.  It  is  seen  that  the  con- 
stant 4  moves  the  curve  4  units  along  the  axis  of  U  and  the  con- 
stant 2  also  moves  it  2  units  along  the  axis  of  X.  Fig.  1.  The 
student  may  form  the  differences  and  test  this.  We  can,  however, 
integrate  this  equation  in  another  way  by  factoring  out  the  con- 
stant 2,  thus 

u  =  J{2x  +  4)  =  J  (X  +  2)2dx  =  (a-  +  2)2  +  C  (18) 

If  X  =  0,  C  =  — 4  which  reduces  this  equation  to  the  pre- 
vious form;  but  if  C  is  taken  as  zero  (for  the  constants  are  gen- 
erally arbitrary)  we  have  the  original  equation  (16).  It  is  thus 
seen  that  integration  may  teach  us  something,  by  giving  us  a 
form  of  equation  we  did  not  previously  know.  And  moreover 
the  works  on  the  Calculus  do  not  generally  explain  this  point 
sufficiently, — why  a  figure  or  quantity  may  be  summarily 
dropped  or  how  a  constant  mysteriously  appears;  But  when 
we  know  that  these  quantities  refer  only  to  the  position  of  the 
curve  upon  the  coordinate  axes,  and  that  the  position  of  the 
axes  is  arbitrary,  then  the  matter  becomes  clear  at  once. 

13.  A  constant  factor  of  the  variable  appears  in  the  differences. 
All  the  differences  are  affected  proportionall\^  The  curve  is 
spread  out  or  contracted. 

In  the  general  equation  9  let  c  =  — -.    Thus 


(19) 


u    = 

1 
2 

X2 

du 

dx  — 

1 
2 

2x  =  X 

d'u 
dx:'  — 

1 

18  An  Introduction  to  the  Differential  Calculus, 


Example  4. 

U  =  V2X 

'.     (19.) 

X 

u 

Ai 

A2 

—4 

-hs.o 

—3.5 

3 

4.5 

2.5 

+1.0 

2 

2.0 

1.5 

1.0 

—1 

0.5 

-0.5 

1.0 

0 

0.0 

+0.5 

1.0 

+1 

0.5 

1.5 

1.0 

2 

2.0 

2.5 

1.0 

3 

4.5 

+1.0 

+3.5 

+4       +8.0 

The  differences  of  this  equation  are  given  in  Example  4.  The 
effect  of  this  constant  factor  is  seen  to  be,  that  all  the  differences 
are  one  half  of  the  corresponding  values  in  Example  1 ;  that  is  in 
the  words  of  the  calculus  **a  constant  factor  appears  in  the 
differentials."  (Courteney)  The  curve  of  this  last  equation  is 
shown  in  Fig.  1  by  a  series  of  dots  upon  the  ordinates.  The 
effect  is  to  spread  out  the  curve,  or  flatten  it  toward  the  axis  of 
X.  If  the  constant  factor  had  not  been  a  fraction,  but  an  integer 
2,  3,  etc.,  the  curve  would  have  been  compressed,— the  branches 
drawn  together. 

14.  A  change  in  the  value  of  the  independent  variable,  alters 
the  interval  of  the  computed  values.  We  will  resume  equations 
10  and  11  of  the  first  example  namely: 

u 


dh 


(20) 


We  also  have  vJ  =  2 

dx" 

As  the  value  of  c/x  is  arbitrary,  it  is  usually  as  a  matter  of 
simplicity  taken  as  unity;  or  in  other  words  whatever  the  actual 
interval  may  be  it  is  considered  as  a  unit.  This  unit  may  be  one 
day  or  one  hour  or  a  period  of  20  days,  etc.  When  interpolating 
a  function  the  interval  of  computed  dates  is  always  considered  as 
a  unit.    In  this  example,  however,  we  will  take  this  interval  as 

one  half,  that  is,  c/x=  -p-. 

Then  the  above  equations  become 

du  —  2xdx       =  2x  .     -7T 


d'a  =  2r/A'2        ^  2   .  (^)^  =V2 


(21) 


Particular  Values  of  an  Equation. 


19 


These  equations  are  shown  in  Example  5. 
Example  5.    Equation  (21). 


1 

V 

=  x2 

du  —    ^ 

. 

X 

u 

Ai 

A  2 

—2 

+4.00 

—1.75 

1.5 

2.25 

1.25 

+  0.50 

1.0 

1.00 

0.75 

+  0.50 

—0.5 

0.25 

—0.25 

0.50 

0.0 

0.00 

+0  25 

0.50 

+0.5 

0.25 

0.75 

0.50 

1.0 

1.00 

1.25 

0.50 

1.5 

2.25 

1.75 

+0.50 

+2.0        +4.00 
If  we  should  take  the  interval  as  one-fourth,   then   we  should 


have 


du  11 

-^  =  '^^  dx=2x.^^^  X 

d^u  ^  11 

^.  =  2dx^    =2.     ^-^ 


(22: 


15.  It  is  seen  from  these  equations  that  by  diminishing  the  in- 
terval of  the  computed  dates  that  the  first  difference  is  diminished 

—  the  second  differences  (  —  )  the  third  differences  (  —  )  and  so  on. 
n  \n  J       ^  \n  J 

In  any  computation  therefore  if  we  find  there  are  higher  differ- 
ences that  are  inconveniently  large  for  interpolation  or  other 
purposes  we  can  cause  them  to  become  less  or  practically  to 
vanish  by  diminishing  the  interval  of  the  computed  quantities. 

16.  Particular  values  of  an  equation.  Maxima  and  Minima. 
The  rule  in  the  calculus  for  this  is  to  differentiate  and  place  the 
first  differential  coefficient  equal  to  zero,  and  solve  the  equation 
with  respect  to  x.  If  .y  results  a  finite  value  or  zero  it  indicates 
one  of  these  two.  This  value  of  x  substituted  in  the  original 
equation,  gives  its  maximum  or  minimum  value.  To  ascertain 
which  one  it  is,  differentiate  a  second  time,  and  if  the  value  of  x 
found  above  makes  the  second  differential  coefficient  positive,  the 
point  is  a  minimum,  but  if  the  sign  results  negative,  the  point  is 
a  maximum. 

To  illustrate  this  we  will  take  equation  (11)  of  Example  1, 
(for  we  have  not  yet  exhausted  this  little  equation.) 


u 
du_ 
dx 
dhi 
dx^ 


—  2x  —  0      Then  x  =z  0 
=;  +  2  positive  value 


(23) 


20  An  Introduction  to  the  Differential  Calculus. 

The  first  differential  coefficient  when  placed  equal  to  zero  and 
solved  gives  x  =  0,  and  this  substituted  in  the  first  of  these 
equations  gives  u  =  0. 

The  second  differential  coefficient  has  the  positive  sign  v^hich 
shows  that  the  value  u  =  0  is  a  minimum. 

This  same  process  can  be  made  use  of  with  the  differences  in 
Example  1.    We  here  look  for    a  point  of  the  curve  which  is 

parallel  to  the  axis  of  X  because  -^  being    the    tangent    of  the 

angle  which  the  curve  makes  with  the  axis  of  X,  if  there  is  a 
minimum  point  the  curve  after  descending  turns  and  then  ascends, 
and  the  lowest  point  is  when  the  curve  is  parallel  to  the  axis  of 
X.  Its  tangent  line  being  parallel,  the  angle  must  be  zero  and 
the  trigonometrical  tangent  is  zero  or  the  first  difference  A^  =  0. 
We  therefore  examine  the  column  of  1st  Differences  for  the  value 
0  opposite  which  we  have  the  values  of  u  and  x.  In  some  equa- 
tions, the  curve  may  approach  the  axis  of  X  and  become  parallel 

(fu 
to  it  but  yet  contain  no  maximum   or  minimum,    -z—^  being  0  is 

neither  +  nor  — .    This  is  the  case  with  the  cubical  parabola 
given  b3^  the  equation  u  —  x?.    The  origin  is  a  point  of  inflexion. 

17.  Points  of  InHexion.    It  was  remarked  in  Article  7  that  the 

d^u 
curve  Fig.  1  was  convex  to  the  axis  of  X  in  which  case  -j-j  is  pos- 

itive.     When  the  curve  is  concave  to  the  axis  -j— ^  is  negative. 

At  the  point  of  inflexion  therefore  -r-j    niust     change    its    sign, 

which  it  cannot  do  unless  it  becomes   0  or  oo.    Hence  these  values 
characterize  a  point  of  inflexion. 

^  =  0  or  ^=oo  (24) 

The  example  in  Article  19  iz  =  x^  shows  a  point  of  inflexion 
when  X  =  0.  A  simpler  example  is  given  in  the  cubical  parabola 
u  ■=  2^  which  may  be  computed,  differenced,  and  plotted,  by  the 
student. 

18.  Cusps.    These  are  indicated  according  to  the  Calculus  by 

-T-  having  two  equal  and  real  values.    Thus  in  the  semi-cubical 
parabola. 

Solving  this  equation  for  various  values  of  x  we  have  the  fol- 
lowing values  and  differences: 


Differential  and  Difference  Distinguished. 


21 


X 

—2 


— 1 


0.000 


=f  1  000 


2.828 


5.196 


8.000 


0.000 
0.000 


1. 000 


1.828 


2.368 


2.804 


=F3.180 


0.000 


1.000 


0.828 


0.540 


0.436 


=F0.376 


A.3 


=F1.000 


:0.172 

0.288 


0.104 


±0.060 


1.172 


:0.116 


=0.184 


=0.044 


5     =F11.180 

There  are  no  real  values  for  the  negative  values  of  x,  and  for 
X  —  0  the  first  real  value  x  —  0  appears;  for  each  positive  value 
of  X  v^e  have  two  real  and  equal  values  with  contrary  signs,— in 
fact  two  curves.  Hence  u  =  0  is  a  cusp-point.  The  form  of  the 
curve,  the  reader  can  doubtless  find  in  any  work  on  the  Calculus. 
\  having  the  double  sign  the  value  u  ==  0  is  a  maximum  for  one 
branch  and  a  minimum  for  the  other. 

19.  The  Distinction  between  a  Differential  and  a  Difference 
rigorously  shown.  Professor  Chauvenet  in  his  Spherical  and 
Practical  Astronomy  has  deduced  a  series  of  formulae  for  trans- 
forming a  series  of  Finite  Differences  into  Differential  coefficients. 
These  formulae  are  similar  to  the  usual  formulae  for  interpola- 
tion, both  being  derived  from  the  algebraic  formula  for  the  sum 
of  the  series.  In  the  present  formulae  in  order  to  interpolate  for 
values  lying  in  a  horizontal  line  with  the  primitive,  the  even 
differences,  and  the  mean  of  the  odd  differences  are  employed; 
but  in  the  general  formula  for  interpolation,  where  the  value  lies 
betw^een  the  given  primitive  and  the  next  following,  the  odd 
differences  and  the  mean  of  the  even  are  employed. 

These  formulae  which  may  also  be  found  in  Watson's  Theo- 
retical Astronomy  are  as  follows,— changing  Chauvenet's  nota- 
tion slightly  to  conform  to  tliat  of  the  present  work.* 


A2   — 


W  \ 
2w 1^/ 

x'  -  W'  (^ 

=^( 

—  w^\^    2 


da 
dx 
cPu 
d 

dx^ 
d^u 
dx^ 
d^u 
dx-"^ 


1       SA; 


6 

1 
12 

1 
4 


A  4  —  etc. 


+ 


1 
30 


2  A. 


etc. 


A  4    +  etc. 


SAs 
o 


+  etc. 


etc. 


(26) 


*  The  author  has  employed  differences  extensively  in  computations  of  the 
planets  for  the  Nautical  Almanac,  U.  S.  Naval  Observatory.  The  hourly  motions 
printed  in  the  Nautical  Almanac,  and  the  aberration  are  computed  by  means  of 
the  first  equation  of  formula  (26).  The  formulae  are  also  used  for  correcting  the 
elements  of  orbit  of  a  planet  from  three  or  more  observations.  In  this  way,  more 
than  twenty-five  years  ago,  he  discovered  this  method  of  explaining  the  Calculus. 


22 


An  Introduction  to  the  Differential  Calculus. 


In  these  equations  the  notation  —;—-  denotes  the  mean  of  the 

adjacent  odd  differences,   lying  above  and   below  the  line  D  in 
Example  6.    The  factor  w  is  used  if  we  wish  to  reduce  the  differ- 
ential to  a  lesser  interval  than  that  of  the  tabulated  values.    As 
we  here  use  the  formulae  it  is  unity,  and  neglected. 
20.    In  this  article  we  will  assume  the  equation 

•  «  =  x-'  (27) 

with  which  we  form  the  following  tabulated  values  with  their 
differences,  Example  6  as  follows: 


] 

Exam 

PLE  6. 

a  =  X*, 

Equation 

27. 

X 

u 

Ai 

As 

A3 

A4 

A5 

-3 

—     243 

+ 

211 

•^2 

— ^32 

31 

-  180 

+  150 

-1 

—    1 

-  30 

—120 

4- 

1 

30 

+120 

0 

0 

0 

0 

+ 

1 

80 

120 

+1 

+     1 

31 

+   30 

150 

+120 

120 

2 

32 

211 

180 

390 

240 

120 

3 

243 

781 

570 

750 

360 

120 

4 

1024 

. 

2101 

1320 

1280 

480 

120 

5 

3125 

2550 

+600 

4651 

-fl830 

+120 

6 

7776 

+4380 

-f9031 

7 

4-16807 

D 


For  the  numerical  values  to  be  used  we  assume 

x  =  ^3 


(28) 


and  for  this  value  the  differential  coefficients  lie  in  the  horizontal 
line  from  3'^  to  D,  while  the  Finite  Differences  lie  in  the  diagonal 
from  3''  to  F. 

The  differential  coefficients  will  now  be  formed,  and  their  values 
for  x  =  +  3  computed,  as  follows: 

da 


dx 

—  ox^, 

d^u 
dx' 

=  20.r^ 

d^u 
dx^ 

=  60x2 

d'u 
dx^ 

=  120x 

d^u 
dx^ 

=  120, 

and  when  x  —  3 


=  405 
=  540 
=  540 
=  360 
=  120 


(29) 


Criticisms  on  Method  by  Limits.  23 

Now  substituting  in  equation  (26)  the  mean  of  the  odd  differ- 
ences, and  the  values  of  the  even  differences  lying  along  the  hori- 
zontal line  D  in  Example  6,  we  have 


du           992 

1 
G 

1140    1 
2   +30- 

240 

-  =  496  —  95  +  4 

=  405 

dx    ~     2 

2 

1 
12 

360 

=  570  —  30 

:=540 

d^u        1140 
dx^            2 

1 
4 

240 
2 

=  570  —  30 

=  540 

f.  -  360 
dx^ 

=  360 

d'"   -120 

dx-" 

=  120 

(30) 


which  values  agree  with  the  differential  coefficients  found  above 
in  equation  29,  by  differentiation. 

21.  It  is  further  and  clearly  seen  that  one  distinction  between 
a  differential  coefficient  and  a  Finite  Difference  is  one  of  Posi- 
tion in  the  scheme  of  tabulated  Finite  Differences.  And  a  Differ- 
ential Coefficient  of  any  order  ma^^  be  correctly  defined  as  that 
certain  interpolated  value  of  a  series  of  tabulated  Finite  Differ- 
ences of  the  same  order,  which  lies  in  a  horizontal  line  with  its 
primitive  function;  while  the  Finite  Differences  are  those  values 
obtained  by  subtraction  which  lie  in  the  downward  diagonal 
from  the  same  primitive  function. 

22.  Further  criticisms  on  the  Method  by  Limits.  We  see  more- 
over from  the  first  equation  of  29  that  when  applied  to  a  function 
of  the  second  power  there  is  but  one  term. 


^ L  V  A  ,   —     j   -^i  (upper  value)  +  V2  A2  o-, . 

xd  —    2    ^  ^^  —     j  -^  (lower  value)—  1/2  A2  ^'^^^ 


which  by  the  differences  of  example  1  when  x  =  3  becomes 

da  1 

Again 


^^^7=    A1-—  A2=7-l  =  6 


u  =  A-2      -^  —  2x,  (when  x  =  3)  =  6 

We  now  discover  what  becomes  of  the  term  h  in  equation      (4) 

171  —  u  —  2xh  +  A^ 

in  which  h  must  be  taken  equal  to  unity  to  correspond  with  the 
interval  in  Example  1.    It  does  not   "disappear"   by  becoming 

less  and  less,  but  is  controlled  or  cancelled  by  the  term  — -—  \  in 

equation  (26)  which  is  1  in  equations  of  the  second  degree,    (pro- 
vided the  coefficient  of  x  in  the  primitive  equation  is  unity.) 
23.     We  are  now  prepared  to  say  that  in  such  equations  as  (7) 


24  An  Introduction  to  the  Differential  Calculus. 

which  we  have  been  considering,  that  for  any  given  value  of  x 
both  u^  —  u  and  2x  are  constants,  (with  coefficients  however,) 
which  becomes  apparent  by  examining  the  changes  which  take 
place  while  h  is  supposed  to  decrease  without  limit.  We  will 
examine  both  equations  (4)  and  (7)  as  follows: 

Equation  (4).  Equation  (7). 

m  —  u=  2xh  4-  A2  "^  ~  "     =  2x  +  h 

(m  -  u)  =2a'+    1 

1 

2  (ui  -  «)  =2a'  +  -^ 


'henx=  1, 

ui  —  u=   2x  -t-    1 

1 
~  2    ' 

2x          1 

1 
~    4 

2x  1 
'  "1-"-    4    +16 

1 
—    8    ■ 

2x  1 
'  "1-"-    8    -^64 

Limit,      0, 

ui  —  u=    0  +    0 

4(ui  — u)  =  2x-\--g 


8  («i  -u)  =2.Y4--g- 
oo(ui  —  u)  ^  2x  -\-   0. 


These  equations  are  convertible  to  one  another  as  they  should 
be;  in  the  first  the  quantity  2x  does  not  remain  to  reach  the  limit 
as  fondly  hoped  in  this  method;  but  becomes  equal  to  zero;  the 
term  (wj  —  u)  also  becomes  equal  to  zero  and  there  is  nothing  left 
but  the  primitive  equation.  This  is  the  correct  and  legitimate 
effect  of  diminishing  the  value  of  /z.  In  the  other  equation  al- 
though the  quantity  2x  remains  yet  the  coefficient  of  the  left 
member  becomes  infinite,  and  at  the  limit  (u^  —  u)   takes  the  in- 

2x 
determinate  form ;  an  inconsistency  arising  from    neglecting 

CX) 

the  fundamental  principle  of  Finite  Differences,  that  h  must  be  a 
constant. 

24.  In  Haddon's  ''Examples  in  the  Differential  Calculus," 
Weales  Series  London,  the  above  mentioned  inconsistencies  are 
very  clearly  shown.     Assuming  the  equation   u  =  j^  the  author 

proceeds,  "— ^ —  =  Sx^  -\-  Sxh  -\-  h^  —   Ratio    of  increment    of 

function  to  increment  of  variable.  Now  the  first  term  of  this 
expression  for  the  ratio  being  3a:^,  it  is  obvious  that  h  may 
undergo  any  change  of  value  whatever,  without  affecting  this 
first  term.  Let  h  then  continually  decrease  in  value  until 
it  is  =  0." 

May  it?  Most  assuredly  it  may;  but  this  sophistry  keeps  out 
of  sight  that  the  first  member  becomes  infinite  from  which  noth- 
ing can  then  be  proved. 

We  see  that  in  the  first  member  of  all  these   similar  equations, 


Finite  Differences  from  Differential  Coefficients.  25 

that  (»!  —  u)  does  not  suffer  much  decrease  in  value.  In  fact  it 
cannot  decrease  smaller  than  the  first  term  of  the  second  member, 
while  its  denominator  h  decreases  without  limit. 

There  seems  to  be  also  another  peculiarity  in  these  equations. 
They  start  with  x  as  a  variable  receiving  the  increment  h,  but 
when  (i/j  —  u)  is  reached  x  seems  to  be  regarded  as  a  constant 
with  h  a  variable ! 

The  true  nature  of  the  change  which  this  method  by  limits 
endeavors  to  explain  is  shown  in  Equations  (26)  to  (30)  where 
it  is  seen  that  all  the  terms  of  the  higher  differences  combine  in 
certain  proportions  by  interpolation  and  are  numerically  added 
to  or  subtracted  from  the  first  term  (which  remains  unchanged) 
and  thus  make  the  change  in  the  first  member  of  the  equation, 
while  this  new  numerical  value  is  still  denoted  by  the  quantity 
3a'2  in  the  present  equations  and  2x  in  those  on  the  previous 
pages.  This  is  an  intricate  point  and  sophistry  can  make  much 
out  of  it.  In  one  sense  3x^  is  constant  and  in  another  sense  it 
has  changed  its  numerical  value.  The  numerical  value  depends 
upon  whether  we  regard  3x^  as  a  term  of  a  Finite  Difference  or  a 
Differential  Coefficient.  How  easy  to  slide  from  one  to  the  other! 
not  knowing  or  suppressing  the  fact  that  the  numerical  values 
are  different. 

Finally.  If  we  have  a  quantity  x  and  add  to  it  another 
quantity  h  and  let  h  diminish  until  it  becomes  zero,  what  in  the 
name  of  common  sense  have  v\re  left  but  the  original  quantity? 

25.  The  reverse  problem.  Finite  Differences  derived  from  Differ- 
ential Coefficients.  Professor  George  Boole  has  given  in  his 
Calculus  of  Finite  Differences*  the  following  formula  which  is  the 
fundamental  relation  between  finite  differences  and  differential 
coefficients: 


dx^  f  1.2.(«+1)  •      dx'^  +  1  [ 

^n  Qn  +  2  d^   +  "^  U  (  (32A) 


"^  1.2.(/2+2)  •      dx"  ^  2     +etc. 


( 


in  which  the  coefl&cients  are  complicated  quantities,— the  numera- 
tors being  computed  by  the  expression 

^      ,     /7  (n  — l)(n  — 2)™           «(/7— l)(w— 2)(/J— 3)"' 
A°0"=n™-/2(/2-l)-  +  — ^ 1^72 -       1  .  2    3 ^-+etc.(32B) 

I  understand  that  in  some  editions  of  Boole's  work  the  numera- 
tors have  been  computed  and  tabulaited.  Applied  to  the  differ- 
entials these  formulae  become: 


Macmillan  &  Co.,  London,  1872. 


26  An  Introduction  to  the  Differential  Calculus. 


du  1       (fu             I       (Pu  1       d^u  1       d^u 

^^—    dx    +  2       dx^    +6       djc3    +  24      dx^    +    120     dx"^ 

_     C?2u  (f'^U                7        f/%                1  cf'^u 

•^2  —  ~1^  +  cfx8  +     12    ~d^  +      4  "c/a'^~ 

_  .^^^     I       3       c?4»  5       d^u 

^^~~djc^     ■       2       c/a'4     +      4      "^^^ 


As 


^^=^]^  +  2-^x^ 


(33) 


dx^ 

Inserting  the  values  of  the  differential  coefficients  from  Equa- 
tion (30) 

Ai  =  405  -r  270  +    90  r  15  +  1  ==     781  ) 

^2  =  540  +  540  +  210  -^30  =  1320 

As  =  540  +  540  +  150   ^  =  1230  }  ^34) 

A4  =  360  +  240  —     600 

As  =120  =     120  J 

These  differences  agree  with  those  in  Example  6  lying  on  the 
diagonal  line  ending  at  F. 

26.  According  to  Boole  the  equation  for  the  several  differences 
in  terms  of  the  Differential  coefficients,  equation  (32)  is  the 
fundamental  relation  between  Differences  and  Differentials.  From 
equation  (33)  it  is  again  seen  that  the  first  differential  coefficient 
is  not  the  same  as  the  First  Difference  which  is  perhaps  the  chief 
fallacy  in  the  usual  explanation  of  the  calculus. 

The  only  exception  to  this  is  that  the  nth  Difterential  coefficient 
is  alwaj^s  equal  to  the  nth  Difference  which  is  a  constant  in  both 
finite  difference  and  differentials. 

27.  Inversion  of  the  Series,  in  Example  6,  Article  20. 

It  will  be  interesting  to  note  what  effect  this  inversion  has  upon 
the  results  of  the  two  formulae  in  Articles  19  and  25.  This  is 
equivalent  to  taking  equation  27  with  the  negative  sign. 

The  Tabular  differences  then  are  as  follows: 


D' 


A- 

4 

+1024 

Ai 
-781 

A2 

+  1320 

As 
-750 

A4 

+480 

As 

-120 

3 

243 

211 

570 

390 

360 

120 

2 

32 

31 

180 

loO 

240 

120 

1 

1 

-  1 

30 

-  30 

+  120 

-120 

0 

+    0 

+    0 

+   0 

The  Differential  Coefficients  still  lie  in  the  horizontal  line  3  —  D' 
while  the  Finite  Differences  are  again  in  the  downward  diagonal 
3  —  F'  which  now  includes  quite  a  different  set  of  numbers  from 
the  previous  table.  Finite  differences  thus  change  when  a  ,  series 
is  inverted.     Changes  of  signs  are  also  noted  above. 


Concluding  Remarks.  27 

By  formula  26  we  find  the  Differentials  from  the  above  Differ- 
ences as  follows: 


du  992 


+ 


dx     —  2^6 

d^u 


1140 
2 

-3^  120=  -  496  +  95- 

4  =  —  405 

80 

4  540 

30 

=  —  570  +30 

=  —540 
=  +  860 
—  -   120 

^(35) 


dx^  =-    ^^0    - 

d^a  _         1140 

dx^  -'~      2     "^ 
d^u 

-d^  =+    360 

d'^u 

Agreeing  numerically  with  those  previously  found  in  the  same 
manner  and  with  the  change  of  signs  the  inversion  of  the  series 
necessitates,  and  also  lying  in  the  horizontal  line  of  the  tabulated 
differences. 

By  formula  33  we  have  the  Finite  Differences  from  the  Differ- 
ential Coefficients,  employing  those  just  found. 

Ai  =  —  405  +  270  —  90  +  15  -  1  ^  —  496  +  285  =  —  211  ] 
A2  =  •-[-  540  —  540  +  210  —  30    = -\-  750  —  570  =  +  180 
^3—— 540  +  540— 150  =  — 150>   /orn 

A4  =  +  360  —  'J40  =  +  120    ^''''^ 

A5=:  — 120  =—120  J 

Corresponding  exactly  with  the  new  values  of  Finite  Differences, 
lying  in  the  diagonal  line  3-F'.  These  two  diagonal  lines  of 
Finite  Differences  are  symmttrically  disposed  above  and  below 
the  horizontal  line  of  Differential  Coefficients. 

28.  Concluding-  Remarks  on  the  Method  by  Limits.  The  fore- 
going pages  following  example  6  Articles  19-20  show  that  the 
differential  coefficients  depend  upon  the  interval  of  the  argument 
x  of  finite  differences  being  taken  as  unity;  whatever  the  actual 
interval  may  be.  We  may,  however,  decrease  the  actwa/ interval 
(Art.  15)  while  still  regarding  it  as  a  unit,  or  even  consider  it 
infinitely  small  as  customary  in  investigations.  The  author  does 
not  wish. to  be  considered  as  holding  that  a  differential  must  be 
unity  or  a  large  quantity.  His  object  being  only  to  prove  that 
h  in  equations  2,  3,  4,  5  must  be  a  unit  and  cannot  be  considered 
as  a  variable  decreasing  without  limit. 

Finally,  the  author  asks  if  the  method  by  limits  can  show  the 
second  differential  coefficient?  Can  it  be  derived  by  another  in- 
finitel3'  small  quantity  k?  Does  its  derivation  follow  the  same 
law  as  the  first  ?  It  would  be  illogical  to  infer  that  it  does,  for 
that  is  begging  the  question — the  very  point  w^e  wish  to  know. 

The  method  by  Limits  regards  a  differential  coefficient  as  a 
ratio  to  which  objection  has  been  made;  but  finite  differences  be- 
ing formed  by  subtraction,  the  differential  coefficient  derived 
from  them  is  not  a  ratio. 


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